PrimeQ[DifferenceRoot[
Function[{[FormalY], [FormalN]}, {-[FormalY][[FormalN]] -
[FormalY][1 + [FormalN]] + [FormalY][3 + [FormalN]] ==
0, [FormalY][0] == 1, [FormalY][1] == 1, [FormalY][2] ==
1}]][727734]]
returns True in Mathematica

To be completed...